# An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(1 ,4 )# to #(5 ,1 )# and the triangle's area is #15 #, what are the possible coordinates of the triangle's third corner?

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The two vertices form a base of length 5, so the altitude must be 6 to get area 15. The foot is the midpoint of the points, and six units in either perpendicular direction gives

Pro tip: Try to stick to the convention of small letters for triangle sides and capitals for triangle vertices.

We're given two points and an area of an isosceles triangle. The two points make the base,

The foot

The direction vector from between the points is

Since the area

So we need to move

Check:

The signed area is then half the cross product

That's the end, but let's generalize the answer a bit. Let's forget about it being isosceles. If we have C(x,y), the area is given by the shoelace formula:

The area is

So if the vertex C is on either of those two parallel lines, we'll have a triangle of area 15.

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The possible coordinates of the triangle's third corner are (3, 1) and (7, 1).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- A triangle has corners at #(9 ,3 )#, #(2 ,5 )#, and #(3 ,4 )#. How far is the triangle's centroid from the origin?
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- Given two ordered pairs (1,-2) and (3,-8), what is the equation of the line in slope-intercept form?

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